preprint - wordpress.com€¦ · ment of individuals (i.e. animals, humans) and their collective...

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A context-sensitive correlated random walk:A new simulation model for movement

Sean C. Ahearna∗, Somayeh Dodgeb,d∗, Achara Simcharoenc, Glenn Xavierd,James L.D. Smithb

International Journal of Geographical Information Sciencedoi:10.1080/13658816.2016.1224887

(in press, August 2016)

aCity University of New York – Hunter CollegebUniversity of Minnesota, Twin Cities

cDepartment of National Parks, Plant Conservation, ThailanddUniversity of Colorado, Colorado Springs

Abstract

Computational Movement Analysis focuses on the characterization of thetrajectory of individuals across space and time. Various analytic techniques,including but not limited to random walks, brownian motion models, and stepselection functions have been used for modeling movement. These fall underthe rubric of signal models which are divided into deterministic and stochasticmodels. The difficulty of applying these models to the movement of dynamicobjects (e.g. animals, humans, vehicles) is that the spatiotemporal signal pro-duced by their trajectories a complex composite that is influenced by the ge-ography through which they move (i.e. the network or the physiography of theterrain), their behavioral state (i.e. hungry, going to work, shopping, tourism,etc.), and their interactions with other individuals. This signal reflects multi-ple scales of behavior from the local choices to the global objectives that drivemovement. In this research we propose a stochastic simulation model that in-corporates contextual factors (i.e. environmental conditions) that affect localchoices along its movement trajectory. We show how actual GPS observationscan be used to parameterize movement and validate movement models, andargue that incorporating context is essential in modeling movement.

keywords: Movement model; stochastic models, agent-based simulation; en-vironmental context; behavior; movement pattern, scale.

∗Corresponding authors. Email: [email protected], [email protected]

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A context-sensitive correlated random walk Ahearn et al., IJGIS 2016

1 Introduction

Movement is essential to almost all organisms. The advent of inexpensive and ubiqui-tous positioning technologies such as Global Positioning Systems (GPS) has resultedin unprecedented datasets that can be used for quantitative assessment of the move-ment of individuals (i.e. animals, humans) and their collective dynamics (Galton,2005). As a consequence, the study of movement has gained significant momentumin Geographic Information Science (GIScience) and its applications in movement ecol-ogy (Demsar et al., 2015; Holyoak et al., 2008), mobility and transportation (Tribbyet al., 2016; Song et al., 2015), behavioral studies (Gonzalez et al., 2008; Sang et al.,2011; Torrens et al., 2012), and public health (Glasgow et al., 2014; Lu and Fang,2014), to name but a few.

Numerous deterministic and stochastic methods have been developed to modelmovement, generate synthetic trajectories, and analyze patterns of movement (Schicket al., 2008; Laube, 2014; Dodge et al., 2016). These methods include various exten-sions of time geography approaches (Winter and Yin, 2010; Hornsby and Egenhofer,2002; Miller, 1991), Levy flights (Jiang et al., 2009; Rhee et al., 2011), random walks(Ahearn et al., 2001; Batty et al., 2003; Gautestad and Mysterud, 2005; Codling et al.,2008; Technitis et al., 2014), brownian bridges (Horne et al., 2007; Kranstauber et al.,2012; Buchin et al., 2012), Step Selection Functions (Squires et al., 2013; Thurfjellet al., 2014), and hidden Markov models (Franke et al., 2004). These models mainlyemploy information about movement parameters (e.g. speed, distance, turn angle)and time to simulate possible trajectories in space and time (i.e random walks andLevy flights), as well as to quantify a probable space accessible or used by dynamicobjects (i.e. time geography and brownian bridge models). Most studies concentrateon the pattern of space use and visit probability (Downs and Horner, 2009; Kie et al.,2010; Benhamou and Riotte-Lambert, 2012; Song and Miller, 2014) while others haveexamined the complex interaction between movement behavior and variation in envi-ronmental conditions (Ovaskainen, 2004; Morales et al., 2005; Moorcroft et al., 2006;Forester et al., 2007; Ovaskainen et al., 2008; Ahearn et al., 2010).

While significant progress has been made, a key question with respect to existingmodels is how well do they capture: the local choices that agents make as they movethrough their environment, their interaction with other individuals (Yuan and Nara,2015), and their own geographic strategies for resource usage (Ahearn and Smith,2005; Dodge et al., 2014; Dodge, 2016). The spatiotemporal signal produced bymovement trajectories is in fact a complex composite that reflects multiple scales ofbehavior from the local choices to the global objectives that drive movement. Move-ment encapsulates several components: the internal states and behavioral states ofthe moving entity, and the space and context (i.e. environmental factors) throughwhich it moves in time (Dodge, 2016). Existing approaches often confound movementpatterns associated with external factors with those patterns associated with globalscale behaviors. Accurate simulation models for movement should incorporate anddeconstruct the environmental and behavioral drivers of movement for more reliable

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and realistic representations. Simulation of movement is essential for studying andpredicting patterns and behavioral responses of moving agents to varying conditionsand their interactions with other individuals.

This study introduces a context-sensitive simulation model for movement based onthe concept of the correlated random walk (Codling et al., 2008). The proposed modelis different from existing approaches because it integrates environmental context intomodeling movement as an integral part of each local choice along the movement tra-jectory. The strength of our approach is that the model is parameterized and validatedusing real movement data. The parametrization of the model is achieved through thedevelopment of probability distributions that relate actual movement to its context.We assess our proposed model on movement of endangered tigers (Panthera tigris)in West-central Thailand. The case study investigates the influence of geography andphysiography (i.e. the shape of home range and landscape characteristics) on tiger’smovement characteristics.

2 Related Work on Modeling Movement

Random walks have broadly been used to model change or movement at local andglobal scales (David and Perry, 2013). A two dimensional or spatial random walkdescribes the probability of moving in a direction from the current position in space.Codling et al. (2008) provide a comprehensive review of different random walk modelsapplied in biology and ecology applications. The simplest version of random walk iscalled the uncorrelated random walk. In this model, the next direction of movementis independent of the direction of the previous movement step. In contrast, thecorrelated random walk (CRW) involves a persistence in the direction of successivemovement steps by introducing a correlation to the preceding directions (Turchin,1998).

Previous studies suggest that the movement of organisms is less random and CRWsare better suited to represent movement (David and Perry, 2013). In addition to thelocal persistence in direction, random walks can involve an external bias to maintaina global direction of movement from an origin towards a destination (Ahearn et al.,2001; Technitis et al., 2014). Random walks form the building block of many agent-based simulation models for movement (Ahearn et al., 2001; Batty et al., 2003;Tang and Bennett, 2010; Torrens et al., 2012; Technitis et al., 2014). Ahearn et al.(2001) built an agent-based model to simulate the collective dynamics of male andfemale tigers (with or without their cubs), and their interactions with prey. Theirsimulation is based on a correlated random walk by introducing external biases (i.e.location of other tigers and location of prey) that are dynamic and a function of theanimal’s state. Gautestad and Mysterud (2005) proposed a multi-scale random walk(MSR) which does not make the assumption of a low-order Markovian process (i.e.the current state is a function of the immediate previous state) that many modelsdo. Their work uses the MSR to model the multiple spatiotemporal scales over whichanimals operate (Gautestad and Mysterud, 2010). In another study, Technitis et al.

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(2014) introduced a point to point correlated random walk model (i.e. for givenorigin and destination locations) based on a set of assumptions on speed capacitiesof moving individuals and a time budget, akin to time geography approaches.

Random walk models are often used to generate a trajectory (i.e. sequence of loca-tions over time). In contrast, time geography and brownian bridge movement modelsmainly quantify a continuos space accessible by a moving individual. As such thesemodels can represent space utilization distribution or visitation probability in spaceand/or in time. Time geography models are based on the concept of Hagerstrand’sspace-time prism (Hagerstrand, 1970). In time geography, accessibility is defined asa space that a moving entity can possibly reach given a time budget and a maximumspeed (Miller, 1991). A variety of time geography approaches have been proposed tomodel movement in a planar space (Winter and Yin, 2010; Song and Miller, 2014),or in a network space (Song et al., 2015). The brownian bridge model quantifies theprobability of being in a location between a start (A) and an endpoint (B) takinginto account the distance between the points and the overall time that it takes totraverse from A to B (Horne et al., 2007; Kranstauber et al., 2012).

Most existing simulation models are parametrized using a set of assumptions andrules based on the dynamic capacities of the moving agents (e.g. speed capacities, timebudget, turn angle patterns). It is now possible to calibrate models based on the dis-tribution functions and correlations of movement parameters (e.g. speed, turn angle)computed from real observations, thus obviating the need for such assumptions. An-other limitation of existing movement models is that many of them ignore the internalstates, behaviors, and context (i.e. external factors) that results in a set of movementpatterns (Dodge, 2016; Dodge et al., 2016). They mainly consider dynamic capacitiesof moving individuals and external biases such as destination or interactions betweenmoving individuals (Miller, 2015). A number of researchers have begun to incorpo-rate context into their understanding and modeling of movement (Ovaskainen, 2004;Morales et al., 2005; Moorcroft et al., 2006; Forester et al., 2007; Ovaskainen et al.,2008). Ovaskainen (2004) and Ovaskainen et al. (2008) used a segmented landscapewith different diffusion coefficients for each landscape type with separate specificationfor the boundary condition. Their research provides a strong case for understandingmovement in heterogeneous landscapes. Moorcroft et al. (2006) proposed a mecha-nistic home range model for carnivores that incorporates conspecific avoidance, roughterrain avoidance and habitat selection to derive home range patterns. The modelsresulting probability density functions were tested against the home range derivedfrom observations of coyote GPS locations and demonstrated a strong “goodness offit”. Schick et al. (2008) provide an excellent summary of these models and otherexisting approaches for modeling animal movement. The models they reviewed focuson three areas: simulating realistic movement, understanding organism-environmentinteraction and its effect on movement, and inferential models for predicting move-ment where data may be incomplete. In conclusion they find that while models haveattempted to understand organism-environment interaction, “none of these modelshas an ability to test for how landscape features actually influence the movement

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process” (Schick et al., 2008). They propose a conceptual model that explicitly in-corporates state, location, and habitat suitability as factors governing the choice ofan individual as it moves from place to place.

A more recent development in understanding the relationship between environ-ment and movement is the Step Selection Function (SSF) (Thurfjell et al., 2014). Themechanism for this model is the Resource Selection Function (RSF) which usually usesa Logistic Regression to relate the probability of selecting the next resource unit toa set of environmental correlates defined by their frequency of use and availability(Thurfjell et al., 2014). While this is a powerful new method to understand theserelationships, it has been used in a limited way to simulate movement. For instance,it has been applied to create a probability surface of use for each resource unit tocalculate a least-cost path to generate point to point movement (Squires et al., 2013).This in essence is a deterministic model of movement in contrast to the stochasticmodel we propose in this paper.

3 Methodology

This section presents a new context-sensitive correlated random walk (CsCRW) methodand compares it with a standard correlated random walk (CRW) to better understandthe contribution of geography and contextual factors to an individual’s trajectory.The models are implemented in a Monte Carlo agent-based simulation environmentin which trajectories are simulated within an area of interest (e.g. defined by theshape of a home range (Powell and Mitchell, 2012)). The simulated trajectories areused to generate a visitation probability surface for evaluation and comparison of themodels, as described below.

3.1 Drivers of Movement

Movement occurs in response to the internal state of a moving agent (e.g. the physi-ological state of being hungry) which results in a change in its behavioral states (e.g.hunting, patrolling) (Ahearn and Smith, 2005). The behavioral states of a movingagent are realized as patterns of movement that occur at different spatial and tempo-ral scales (Dodge, 2016). Environmental factors (e.g. slope, vegetation density, preydensity and the existence of trails) often affect local scale choices in movement (e.g.direction and rate of movement). These choices are made based on local conditionsunder the broader constraints of more global scale behaviors driven by internal statesof the moving agent (Figure 1). In this paper, context refers to environmental fac-tors that may influence the movement of an agent at local scales. Understanding thecontribution of individual’s behavior at different scales to the observable patterns ofmovement is critical for their proper interpretation. In this paper we focus on thelocal scale choices affected by environmental context; how to calibrate them and howto validate their importance.

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internal state(e.g. biological state)

behavioral state(e.g. movement goal)

environmental correlates(e.g. slope, vegetation)

local choices

patterns of movement

Figure 1: Drivers of movement

3.2 Correlated random walk (CRW)

The algorithm implemented here (Figure 2) is a modified version of the standardCRW (Turchin, 1998) to simulate a trajectory T (equation 1). The algorithm firstselects a random starting point (Step 1) and picks a persistence value for direction(p) from a random uniform distribution (Step 2). If it is less than the input valueof persistence in direction, as calculated from the actual GPS observations, then theagent persist in the previous direction with a small probability of deviation (e.g.standard deviation 10 degrees, Step 3.a). If it is more, then the agent can move witha much larger turn angle (e.g. standard deviation 45 degrees, Step 3.b) to a newdirection (α, equation 2), to reduce back tracking. Next, a distance d is selected fromthe probability density function derived from GPS observations (Step 4, equation3). The agent is then moved for distance d in direction α forming a vector (Step 5,equation 4). This process is repeated to generate a trajectory in the area of interestusing the same number of points as the actual GPS trajectory (Figure 2). The formalrepresentation of the process is as follows:

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T = {(x0, y0, t0), (x1, y1, t1), .., (xi−1, yi−1, ti−1), (xi, yi, ti), ..., (xn, yn, tn)} (1)

αi = αi−1 +N(0, σ2),where σ =

{10◦ if p ≥ persistence

45◦ otherwise(2)

di = t ∗ χ2(µ),where χ2(µ) is the distribution of speed obtained from GPS(3)

xi,t = d ∗ cosαi + xi−1,t−1 (4)

yi,t = d ∗ sinαi + yi−1,t−1

Highest direction probability is

in the direction of movement

Lowest direction probability

primary movement direction

start

1. pick a random start location in the area of interest

3.a. pick a random direction (a) from a normal distribution

0next turn angle

prob

abili

ty

10-10

4. pick a distance (d) from a distribution

5. move d meters in the selected direction (a)

xi,yi

xj,yj

da

x0,y0

no_pts > MAXno

endyes

2. pick a directionpersistence probability

value (p)

p ≥ persistence yes

probability in each direction

primary movement direction

3.b. pick a random direction (a)

0next turn angle

prob

abili

ty

45-45

no

µ

c c2

Figure 2: Correlated Random Walk (CRW)

3.3 Context-sensitive correlated random walk (CsCRW)

The CsCRW is a modification of the CRW algorithm to account for contextual fac-tors (i.e. slope) in generating a context-sensitive trajectory (T c, equation 5). The

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movement of the agent across space is simulated on the per-pixel basis rather thanby a vector as in the CRW model, to incorporate local choices made by the agentbased on context as it moves through space. Contextual factors are modeled by usinga probability density function P (c) to determine the next probable move (Step 5.1,Figure 3, equation 6). This function is approximated by a chi-square distributionχ2(µ), derived from the relationship between the agent’s locations (i.e. GPS observa-tions) and the environmental context values (i.e slope use). The CsCRW algorithmfollows the same process described for the CRW until Step 4 in Figure 2. Instead ofmoving along a vector in direction α for distance d, the CsCRW moves one pixel ata time based on context (Step 5.1, Figure 3). At each step, the pixel which is in di-rection α and its two adjacent neighbors become possible choices for the agent’s nextmove (Step 5.2). Among the three choices, the pixel whose context value is closest tothe random context variable (c) selected from P (c) is chosen (Step 5.3, equation 7),formally:

T c = {(x0, y0, c0, t0), .., (xi−1, yi−1, ci−1, ti−1), (xi, yi, ci, ti), ..., (xn, yn, cntn)} (5)

P (c) = χ2(µ),where χ2(µ) is distribution of context values used by an agent(6)

ci = Cj|min(|Cj − c|),where j ∈ J = {1, 2, 3} (7)

(xi,t, yi,t) = (X(ci), Y (ci)),where X, Y are the coordinates of choice ci (8)

The agent is moved to the selected pixel (Step 6, equation 8 ) and the processcontinues until the selected distance d is reached. When that happens it returns toStep 2 of CRW (Figure 2). This process is repeated to generate a trajectory in thearea of interest using the same number of points as the actual GPS trajectory.

This approach could in fact include any number of contextual factors (m) thatcan be combined through a joint probability function (P(c) in equation 10), formally:

ci = {c1, c2, ..., cm},where m is the number of contextual factors (9)

P (c) =m∏k=1

P (ck) (10)

3.4 Assessment of models

To assess the CRW and CsCRW models a surface for probability of visitation iscreated for each model in the area of interest (e.g. home range). Each surface iscreated using a Monte Carlo simulation of 1,000 trajectories and intersecting thetrajectories with the raster of the area of interest and accumulating the number ofvisits for each cell. To evaluate how well the surface for probability of visitationcaptures the actual visitation probability of the moving entity, its GPS trajectory iscompared with a random trajectory of equivalent size. Our premise for evaluation isthat the best model will result in higher cell counts along the GPS trajectory, thanfor the random trajectory.

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step 4.,CRW

5.2. take three pixels according to the selected direction from step 3.

5.1. pick a random context variable (c)

µ

5.3. select a pixel with a context value

closest to c (from 5.1.)

6. move to the selected pixel

current pixel (xi,yi)

c c

(xi,yi)

(xj,yj)

2

step 2., CRW

is distance d reached?

no

yes

Figure 3: Context-sensitive Correlated Random Walk (CsCRW)

4 Case Study and Results

4.1 Dataset

To evaluate our proposed methods we use movement data for two female tigers trackedin Huai Kha Kaeng Wildlife Sanctuary in Thailand. Both tigers were fitted witha Vectronic Aerospace GmbH GPS Plus collar. Tiger T7203 was tracked betweenDecember 2009 and July 2010 with a sampling rate of one hour, and tiger T10727 wastracked between December 2012 and January 2015 with a sampling rate of four hours.Following removal of outliers and erroneous GPS observations, T7203 consisted a totalof 4874 points and T10727 consisted of 3946 GPS points. Figure 4 shows trajectoriesand computed home ranges of the two tigers, and the elevation maps of the areas.The home range of the tigers are calculated using both a 100% Minimum ConvexPolygon (MCP) (Mohr, 1947) (shown in magenta in Figure 4) and a CharacteristicHull Polygon (CHP) (Duckham et al., 2008) (shown in dark blue). The characteristichull polygon is generated from the Delaunay triangulation (Okabe et al., 1992) ofthe set of GPS observations. The CHP is extracted from the Delaunay triangulationthrough the removal of triangles determined by the chi-shape algorithm, resultingin a bounding hull with non-convex edges (Duckham et al., 2008). This methodwas selected for the analysis because it is one of the most parsimonious methods fordetermining a home range, as can be seen in the comparison with the commonly used

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MCP approach (Downs and Horner, 2009) (Figure 4). The generated home rangesusing the Delaunay triangulation method resulted in a 34.26 km2 boundary for tigerT7203 (93.7% of MCP), and a 99.92 km2 boundary for T10727 (89.1% of MCP). Asseen in Figure 4, the home ranges of the tigers show a diversity of terrain. Both tigertracks are annotated with elevation and slope information from the digital elevationmodel (DEM) of the study areas obtained from ASTER GDEM dataset1 (30 meterresolution) using a bilinear interpolation (Dodge et al., 2013).

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Thailand

MyanmarVietnam

Cambodia

! Observed GPS PointObserved Trajectory

Elevation (m)

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0 1 20.5 Km

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Thailand

MyanmarVietnam

Cambodia

! Observed GPS PointObserved Trajectory

Elevation (m)

Low : 93

Characteristic Hull Polygon

High : 1946

0 2 41 Km

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(b) T10727 trajectory and home range

Figure 4: The trajectories and home ranges of two female tigers, and the elevationmaps of the study areas.

1Data source: http://asterweb.jpl.nasa.gov/gdem.asp

10

preprint


4.2 Model Parameterization

Three model parameters are calibrated from analysis of the tigers GPS data. Theseparameters are: directional persistence, speed of movement, and slope preference.Directional persistence and speed of movement are used to parametrize both theCRW and CsCRW models, while the slope persistence is used for the CsCRW model,as an example of a context variable. Directional persistence is calculated as the ratioof the total number of turn angles between −20◦ and +20◦ to the total number of GPSvectors and resulted in a value of 0.3. To parametrize speed of movement, a probabilitydensity function is created using 1-hourly GPS observations and estimated by a χ2

distribution with a mean of 0.24 km/hr (Figure 5). To parameterize slope preferenceprobability density functions are created from the slope annotated GPS observationsof the two tigers, and were estimated with χ2(µ = 5.8◦) for T7203 and χ2(µ = 6.1◦)for T10727, as shown in Figure 6. It is important to remark that the data of the twotigers resulted in very similar mean estimates for the χ2 distributions of their slopeuse.

It is necessary to note that since speed and directional persistence are spatial-temporal measures, they are computed only using the 1-hour data set for parame-terizing both models. However, since slope preference is a spatial phenomenon, thetemporal resolution of the data has no effect on estimation of slope use. Thereforethe differing temporal resolutions of the two tigers do not impact the model parame-terization or the results of the validations.

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4.3 Results

4.3.1 Correlated random walk (CRW)

Speed distribution and directional persistence obtained from model parameterizationare used as input to calculate the distance and direction for each successive move inthe correlated random walk model (as described in Section 3.2). Figures 7(a) and 8(a)show probability of visitation surfaces, generated through a Monte Carlo simulationusing a CRW model (as described in Section 3.4), for tigers T7203 and T10727. Theresolution of the generated probability of visitation surfaces is the same as the DEM(30 meters). The highest visitation counts (shown in red) are away from the borders ofthe home ranges and the areas near the borders show lower visitation counts in blue.As seen in the Figures 7(a) and 8(a), this model captures the impact of geography(i.e. location within the home range) on visitation probability.

4.3.2 Context-sensitive correlated random walk (CsCRW)

The sensitivity of tiger movement to slope (i.e. slope preference) for the CsCRWmodel is parametrized for the two tigers using χ2 distributions of µ = 5.8◦ andµ = 6.1◦ as shown in Figure 6. These functions are used to incorporate slope selectionfor each successive move in the CsCRW model (as described in Section 3.3). Figures7(b) and 8(b) show probability of visitation surfaces, generated through a Monte Carlosimulation using a CsCRW model (as described in Section 3.4), for tigers T7203 andT10727. The resolution of the generated probability of visitation surfaces is the sameas the DEM (30 meters). As seen in the figures, this model captures the impactof terrain on the visitation probability of the agent. The areas in blue (i.e. lowvisitation) correspond to the high slope areas or hilltops, which are difficult to reachfor tigers. These rougher terrains, as highlighted on slope maps on Figures 7(c) and8(c), result in the creation of corridors of high visitation for the CsCRW model (i.e.higher visitation counts shown in red) along river bottoms, valleys, and ridge lines.

4.4 Assessment

Two tiger trajectories from GPS observations are used for validation. For each tiger,a random trajectory of the same number of points as the tiger trajectory is generated(i.e. 4874 points for T7203, and 3946 points for T10727). The visitation count foreach pixel along the tiger trajectory and the random trajectory is extracted from theprobability of visitation surfaces of both models (i.e. CRW and CsCRW) as describedin Section 3.4. The results are presented as boxplots in Figure 9 and summarized usingmeans, medians and their differences for the tiger and random trajectories (Table 1).

Both CRW and CsCRW models result in significant higher means and mediansfor both tiger GPS trajectories as compared to the random trajectories, using a T-squared test (p < .0001). The CsCRW model shows much higher mean and mediandifferences for visitation counts (i.e. diff in Table 1) than the CRW model for bothtigers. For T7203, the CsCRW mean difference is 245.4 (visits per cell), while the

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Figure 7: Visitation counts for the home range of tiger T7203 obtained from (a) CRWand (b) CsCRW models; and (c) the slope map for tiger T7203. The area of the homerange is 34.26 km2.

Table 1: Comparative assessment of CRW and CsCRW models for tigers T7203 andT10727.

CRW CsCRWtiger random diff tiger random diff

T7203mean 1002.0 928.5 73.5 1144.0 898.6 245.4median 1056.0 1028.0 28.0 908.0 684.5 223.5

T10727mean 385.7 349.9 35.8 367.7 307.9 59.8median 396.0 378.0 18.0 355.0 307.0 48.0

CRW mean difference is 73.5. For T10727, the CsCRW mean difference is 59.8, whilethe CRW mean difference is 35.8. The CRW model shows a low variance (Figure9) because it captures the relative uniformity in space use by the tiger towards themiddle of its home range. More visits occur toward the center of the home range andaway from the boundary due to the need to cross the middle of the home range whenmoving from one part of the home range to another. However, as seen on Figures 7(a)and 8(a) the model does not reflect the lower space use and movement corridors usedby the tigers in the rougher terrain of the home ranges (illustrated in Figure 4). Incontrast, the CsCRW model has a much higher variance with a pronounced skew abovethe median (Figure 9). This can be attributed to the fact that the CsCRW modelincorporates context (i.e. slope) into the choice of movement and better captures thecorridors which tigers use more frequently. For instance, high visitation corridors,shown by the CsCRW model, in the eastern slopes of T7203 (Figures 7) and in thenorthwestern part of the home range of T10727 (Figure 8), coincide with the actualtiger movement track as seen in Figure 4. The more distinct corridors of modeledsurfaces can also be observed in the more rugged portions of both home ranges. This

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Figure 8: Visitation counts for the home range of tiger T10727 obtained from (a)CRW and (b) CsCRW models; and (c) the slope map for tiger T10727. The area ofthe home range is 99.92 km2.

is not observed in the CRW model for either tiger’s home range. It is important tonote that due to the difference in the size of the two home ranges, and therefore thedifference in the scales of two figures, the influence of micro-topography is seen morepronouncedly in Figure 7 than Figure 8.

5 Discussion

The underlying principle of this paper is that modeling trajectories using geometricmetrics (e.g. path sinuosity, step length, turn angle, etc.) ignores the fact that atrajectory is a complex and composite signal of various behaviors that occur acrossmultiple spatial and temporal scales. The goal of this paper is to demonstrate thatlocal choices (i.e. decisions and behavior at local scales) play a major role in wherea moving agent (e.g. tiger) is likely to go. Two stochastic models are developedand examined, which estimate the space use and movement by an individual. Thesemodels simulate agents to move through space using two algorithms, a correlatedrandom walk (CRW) and a context-sensitive correlated random walk (CsCRW). TheCRW results in a space use that is more pronounced away from the boundary (inthis case the tiger’s home range). It captures the global patterns of space use for anarea by an individual. In contrast, the CsCRW model incorporates the contextualfactors (in this case slope) which influence the strategies for local movements by anindividual.

The premise of our analysis is that local scale effects must be understood andmodeled before regional and global scale behaviors can be deduced from a trajectory.For instance, a tiger that is patrolling its home range boundary (i.e. global scalebehavior), still makes choices at local scales (i.e. following contour lines or movingalong natural trails) to traverse its home range. There may be external biases driving

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tiger random

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Figure 9: Boxplots of visitation counts of CRW and CsCRW models for randomtrajectories and tigers T7203 and T10727

tiger’s global trajectory, but how the tiger reaches its goal-oriented destination isinfluenced by local choices. Therefore we make a distinction between local choicemodels (i.e. CsCRW) and biased CRW. The external biases constrain the possiblepaths to the goal but the local choices are largely independent of theses biases. Otherapproaches such as Step Selection Function (SSF) can also be used to understandthe relationships between local scale environmental correlates and movement choicebased on available resources, however, their use has been confined to deterministicmodels for movement (Squires et al., 2013) in contrast to the stochastic approach wehave taken in this research.

A number of contextual factors play a role in local movement patterns and spaceuse of an individual. The strength of the CsCRW model is that it can incorporatemultiple contextual factors. In the case of the tiger this could include slope, vege-tation density, prey density or proximity to the boundary. Parametrization of thesefactors can be made using actual field observations and GPS locations. In fact, thestrength of our approach is that the models are calibrated and validated through

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actual observations. In the case of the CsCRW model implemented in this researchthe parametrization is done using the tiger’s occurrence on different slope classes. Asimilar process could be used to parametrize other contextual information that con-tributes to tiger movement choices. In this study, we validated our model using twodifferent trajectories of two tigers. These trajectories were different in terms of theirgeography (i.e. geometry of their home ranges), physiography (i.e. terrain type andtopographic features), as well as the temporal resolution of their GPS observations.The comparative analysis of the two home ranges suggests that the proposed simula-tion model successfully incorporates the influence of geography and topography (i.e.context) on space use and local choices made by the tigers.

Movement is driven by an individual’s state and the associated behaviors thatoccur at different spatial and temporal scales (Ahearn et al., 2001; Gautestad andMysterud, 2005). The resulting trajectory of movement is therefore a complex com-posite (i.e. signal) that is influenced by geography through which individuals move(i.e. the network or the physiography of the terrain), their behavioral state (i.e.hungry, going to work, shopping, tourism, etc.) and their interaction with otherindividuals. The study of movement aims at understanding these components andthe scales over which they occur (Dodge, 2016; Dodge et al., 2016). In this researchwe use a spatiotemporal stochastic simulation procedure that incorporates space andenvironmental context as two important components of the signal associated with atrajectory. We see context as a driver for the local choices that individuals may make,that are nested within more global scale objectives driven by the animals behavioralstate. For instance a tiger patrols its home range at certain intervals, however thepath it takes may be influenced by the characteristics of the terrain and vegetation.Thus the pattern of movement is a reflection of both of these factors at two very dif-ferent scales. In essence by modeling these contextual parameters we are separatingthem out from the patterns of movement which are attributed to the agents behaviorsstate.

6 Conclusion and Future Work

The main contribution of this study is to integrate environmental context in a spa-tiotemporal simulation model for movement, and to use real tracking observationsto parameterize and validate the simulation. Three parameters (i.e. directional per-sistence, slope use, speed) were calibrated in this research from GPS tiger trackingdata of two tigers. Two different models (CRW and CsCRW) were developed andevaluated for estimating probability of visitation (or space use) by a tiger within itshome range. The CsCRW model shows the most promise for its ability to incorporatemultiple contextual variables that influence movement choices at local scales in spaceand time.

The spatiotemporal simulation proposed in this research incorporates the threeimportant components of movement: space, time, and context as in the frameworksuggested by Dodge (2016). This research did not model the state and objectives of

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individuals and the resulting higher level behaviors (i.e. patrolling, hunting, tourism,shopping) which drive large scale movement patterns. Future extension of this methodwill incorporate these larger scale behaviors.

Acknowledgments

S. Dodge and G. Xavier work was supported under the 2015 UCCS Committee onResearch and Creative Works (CRCW) and the UCCS College of Letters, Arts, andSciences Student-Faculty Research Awards. We thank Saksit Simcharoen, SomphotDuangchantrasiri, Somporn Pakpein, Onsa Norasarn for assistance in the field. TheThai Olympic Fibre-Cement Co., Ltd. and the USFWS Tiger Rhinoceros Fundfunded the tiger research. The authors wish to thank the reviewers for their insightfuland constructive comments.

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